Animated Solution for Mathematics - Definite Integration: If $f:\mathbb{R}\to \mathbb{R}$ is given by $f(x)=x+1$, then the value of ${\lim }_{n\to \infty }\frac{1}{n}\left[f(0)+f\left(\frac{5}{n}\right)+f\left(\frac{10}{n}\right)+\cdots +f\left(\frac{5(n-1)}{n}\right)\right]$ is:

Imagine you are standing before a massive, intimidating wall of numbers. You see a limit, a summation, and a function $f(x)=x+1$.

It looks like a chaotic mess of terms, a series that stretches to infinity. But in the world of JEE Advanced, this is not a wall; it is a doorway.

This problem is a classic invitation to use the Riemann Sum, a powerful bridge that connects the discrete world of summation to the continuous world of integration. Let us walk through this journey together.

First, let us look at the expression:

It looks daunting, but notice the rhythm. The arguments inside the function are $0,\frac{5}{n},\frac{10}{n},\dots ,\frac{5(n-1)}{n}$.

This is clearly a sequence where the numerator increases by $5$ each time. We can write this compactly using Sigma notation:

Now, the beast is tamed. We have a clear general term.

Here is where the magic happens. The Riemann Sum theorem tells us that a limit of a sum of the form ${\lim }_{n\to \infty }\frac{1}{n}{\sum }_{r=0}^{n-1}g\left(\frac{r}{n}\right)$ is exactly equal to the definite integral ${\int }_0^{1}g(x)dx$.

Think of this as adding up the areas of infinitely many, infinitely thin rectangles. As $n$ approaches infinity, the width of each rectangle, $\frac{1}{n}$, becomes the differential element $dx$, and the discrete index $\frac{r}{n}$ becomes the continuous variable $x$.

We must be precise. Our sum starts at $r=0$, so the lower limit is ${\lim }_{n\to \infty }\frac{0}{n}=0$.

It ends at $r=n-1$, so the upper limit is ${\lim }_{n\to \infty }\frac{n-1}{n}=1$.

Our summation has now transformed into the integral:

This is the heart of the problem. We have moved from a discrete sum to a continuous area.

We know $f(x)=x+1$. Therefore, $f(5x)=5x+1$.

Our integral becomes:

This is a simple linear function. Integrating term by term, the integral of $5x$ is $\frac{5x^2}{2}$, and the integral of $1$ is $x$.

Evaluating this from $0$ to $1$, we get:

Plugging in the upper limit, we get:

Plugging in the lower limit gives us $0$. The final result is $\frac{7}{2}$.

Look at that! The chaos has resolved into a single, elegant fraction. You have successfully navigated the Riemann Bridge. Keep this perspective, and no limit will ever intimidate you again.